Calculus III Exam Review

Calculus III Exam Review

Ashish HedaPeter Zhu

List of Topics11.5• Equation of Line• Equation of Plane• Types of Lines/Planes• Distance (from point to line/plane or from

line to plane)• Line/Angle of Intersection between Planes11.6• Visualize/Name the Equation12.1• Rules for Differentiating Vector Functions• Velocity. Speed, Acceleration12.2• Integration of Vector Functions• Initial Value Problem/Projectile Motion• Rules for Integrating Vector Functions

12.3• Arc Length• Arc Length Parameterization• Speed On Smooth Curve12.4 • Unit Tangent Vector• Normal Vector• Curvature• Osculating Circle & Radius of Curvature12.5• Tangential and Normal Acceleration• Binormal Vector• Torsion12.6• Velocity and Acceleration in Polar CoordinatesPossible Tricky Problems

Equation of LineRecall that in 2D the equation of a line is given by:

where m represents the slope In three dimensions we have a similar equation:

*Note the bolded terms show these are vectors rather than scalar values

Return to List of Topics

Equation of Plane

You need a way of describing every point on a plane. What is something every line on the plane will have in common?

Answer: all lines make a 90° angle with the normal vector of the plane and thus, the dot product of the normal line and any line on the plane is 0.


Continued

Types of LinesReturn to List of Topics

Types of Planes

Distance From Point to LineReturn to List of Topics

Distance From Point to PlaneSimilar to finding distance from point to a line,

except that you are given the slope of the perpendicular line!


How can you find a Point P on the plane?

Use the equation of the plane and set any two variables to 0 and solve for the other:

Line/Angle of Intersection of Two Planes

In order to determine the angle/line of intersection take a look at the following picture:

The normal vectors of two planes can be easily obtained.

What can you say about the angles depicted in the picture to the left?


How can you determine the angle between the two normal lines?

How can you determine the slope of the line of intersection?

Now all you need is a point on the line and you can find the line of intersection.

From here you can find theta and determine the angle between the planes. Remember that there are always 2 possible angles between planes! Usually you are asked to provide the acute angle.

Line of Intersection

Now we need a point. This is any point that satisfies both of the above equations. There are lots of methods for doing this but here is one:

Visualize/Name the Equation

What type of equation is it? What does it look like?

Parabola that goes forever along the x-axis


Cylinder (with elliptical base)

Ellipsoid

Paraboloid

Hyperboloid (two unconnected regions)

Hyperboloid (two connected regions)

Rules for Differentiation of Vector Functions


Velocity, Speed, Acceleration

Problem 23 from section 12.1• Does the particle have constant speed? If so, what is it?• Is the particle’s acceleration vector always orthogonal to

its velocity vector?• Does the particle move clockwise or counterclockwise

around the circle?• Does the particle begin at the point (1,0)?

IntegrationReturn to List of Topics

Initial Value Problems of Vector FunctionsReturn to List of Topics

Projectile MotionA projectile is fired with an initial speed of 500 m/s at an

angle of elevation of 45°• When and how far away will the projectile strike?• How high overhead will the projectile be when it is 5 km

downrange?• What is the greatest height reached by the projectile?First, thing is convert this into a problem like the one before:

Possible information you may need to know

• Acceleration due to gravity in m/s2 and ft/s2

• How many meters in a kilometer (km) …

Rules for Integration of Vector Functions

(see problem 33 of 12.2 for more material)

Note that there is no property listed for a non-constant vector crossed with r(t).


Arc Length Along a Curve Recall that for 2D, the arc length is given by:

Thus, it makes sense that in 3D, the arc length is given by:


Arc Length Parameterization with base point

Sometimes we want to know what is the arc length from an initial position to any given position at time, t.

Ex. Tracking a satellite: you know its initial position where it was put into outer space and you know the path it will follow. You want to design a formula so that at any given time you know how much distance it has traveled


Find the arc length parameter along the curve form the point where t = 0 for the curve:

Then, re-write the curve as function of the arc length parameter.


Speed on Smooth CurveReturn to List of Topics

Unit Tangent Vector

Represents the change in direction of the curve with respect to time or the direction of the velocity at anytime, t.


Normal Vector

This is change of the change in direction of the curve with respect to time or the change in of the direction of velocity with respect to time.


Binormal Vector

Has physics and other applications but it is important for completing the Frenet frame. This is similar to the xyz frame except unlike the xyz frame this frame travels with the curve.


Cuvature

The rate at which the direction of the curve changes with respect to the distance traveled.

Best way to think about curvature is driving a car. If the direction of travel changes from driving north to driving east in 1 mile then it not that difficult of a turn to make. On the other hand going from north to east in a matter of 100ft is much harder and thus a much greater curvature.


You can also think of it in terms of driving along a circle: Imagine you are going at 100 mph around the following two circles:

Harder to drive around

Easier to drive around

R=100 ftR=5,280ft

Thus, curvature can be approximated by:

The actual formula is:

Computational Formula of Curvature

Osculating Circle or Circle of Curvature and the Radius of Curvature

• Tangent to the curve at P (has the same tangent line the curve has)

• Has the same curvature the curve has at P

• Has center located along the direction of the Normal vector.

• Radius of curvature =


Tangential and Normal Components of Acceleration

An easy way to understand this is that the tangential component of a(t) points in the direction of the unit tangent vector (or the velocity vector) and similarly the normal component of a(t) points in the direction of the normal vector.

Torsion

Rate at which the osculating plane turns about T as P moves along the curve. Measures how the curve twists.


Computational formula

Possible Tricky Questions

(True/False)Three statements: I. Two perpendicular lines to a plane are parallelII. III. The derivative of the arc length is the velocity.

(answer: T, T, F (derivative of arc length is speed)(Types of Planes, Rules of differentiation or

integration, acceleration components, arc length, and curvature concepts…)

• May give you 4 points and ask you if all lie on the same plane

• Distance from a line to a plane (be careful to check that the line does not intersect the plane)

• Ask you to re-write a curve with the arc length parameter (this means you should provide r(s))


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Calculus III Exam Review